A Survey on Varieties of PG(4, q) and Baer Subplanes of Translation Planes

Abstract

Publisher Summary This chapter presents a survey on varieties of PG(4,q) and Baer subplanes of translation planes. It is known that a translation plane π of dimension 2 over its kernel F=GF(q) can be represented by a projective spacer Σ 4-dimensional over F, fixing a hyperplane Σ' and a spread S of lines of Σ'. The points of π are represented by: (i) the points of Σ- Σ', and (ii) the lines of S. The lines of π are represented by: (i) the planes α of Σ - Σ' such that α∩Σ' belongs to S and (ii) the spread S. The translation line of π, say it l, is represented by s. It is known that the planes β of Σ- Σ' such that β ∩ Σ' does not belong to S (such planes will be called “transversal planes”) represent the Baer subplanes B of π such that B ∩ l is a line of B—that is, the so called “affine Baer subplane.”